define the prop of a driver with a script?
Firstly, quite possibly, the issue re viewport via render is you need to drive the hide property as well.  Simply right click copy on view_render driver, and right-click paste driver into the hide field to emulate hide_render behaviour.

A driver can only return a float (decimal) or integer (or bool (True, False)) type, therefore you can't return object references or name strings.  What you can do though, is pass the name of an object to a driver expression.

To get a property from a parent using a driver.

def get_parent_property(name, property, default):
    object = bpy.data.objects.get(name)
    parent = object.parent
    return getattr(parent, property, default)

#add to driver namespace
bpy.app.driver_namespace["get_parent_property"] = get_parent_property


The driver expression get_parent_property("Cube", "hide_render", True) will return the hide_render property of "Cube"'s parent object, or the default value True in the case there is no parent.